LV SURFACE RECONSTRUCTION FROM SPARSE TMRIUSING LAPLACIAN SURFACE DEFORMATION AND OPTIMIZATION

Shaoting Zhang, Xiaoxu Wang, Dimitris Metaxas

Rutgers, the State University of New JerseyComputer Science Department

Ting Chen, Leon Axel

New York UniversityRadiology Department

ABSTRACT

We propose a novel framework to reconstruct the left ventri-cle (LV)’s 3D surface from sparse tagged-MRI (tMRI). Firstwe acquire an initial surface mesh from a dense tMRI. Thenlandmarks are calculated both on contours of a specific newtMRI data and on corresponding slices of the initial mesh.Next, we employ several filters including global deformation,local deformation and remeshing to deform the initial surfacemesh to the image data. This step integrates Polar Decom-position, Laplacian Surface Optimization (LSO) and Defor-mation (LSD) algorithms. The resulting mesh represents thereconstructed surface of the image data. Further more, thishigh quality surface mesh can be adopted by most deformablemodels. Using tagging line information, these models canreconstruct LV motion. The experimental results show thatcompared to Thin Plate Spline (TPS) our algorithm is rela-tively fast, the shape represents image data better and the tri-angle quality is more suitable for deformable model.

Index Terms— Reconstruction, deformable model, lapla-cian surface, optimization, triangle quality, remeshing

1. INTRODUCTION

tMRI is a non-invasive way to track the in vivo myocardialmotion during cardiac cycles. Reconstructing the 3D LV mo-tion from tMRI can assist doctors to diagnose cardiac diseasesearlier, and can also be used for 3D strain analysis of the my-ocardium. A typical approach of reconstruction is to calcu-late the motion of material markers from tags. Employingthese material markers as control points, deformable modelscan reconstruct myocardial motion. Mesh based models ap-proaches such as Finite Element Methods (FEM) and quadricmodels have been widely used for LV motion reconstruction.An accurate and high quality mesh is crucial to initialize thesemodels. If the tMRI data is dense enough, the resulting meshcan be acquired by segmentation and triangulation. However,in many cases dense data is unavailable. An alternative ap-proach is to obtain a high quality mesh from a generic densetMRI, and then deform this mesh to any sparse data for sur-face reconstruction.

In our previous research [1], a plausible mesh is built bymanual segmentation with validation by an expert, and Delau-nay triangulation using geodesic distances. Next landmarksare calculated from this mesh and any sparse LV tMRI. Fi-nally TPS [2] are employed to deform this mesh to LV tMRI.Due to TPS, the whole procedure is slow and the resultingmesh is not good enough measured by radius ratio.

This paper presents an effective deformation method,which can replace TPS. The algorithm consists of global de-formation, local deformation and remeshing. LSO [3] andLSD [4] are employed for remeshing and local deformationrespectively. The output is the reconstructed surface meshand it can be employed as the initial mesh for deformablemodels. For modeling purpose, this mesh should be accurateand the triangle quality should be good [3]. Our output meshis tested with different deformable models, such as FEM [1][5], Laplacian Volume Editing [6], volume registration basedtracking [7] and mass-spring based method [8]. The resultsshow that it is suitable.

Our paper is organized as follows: section 2 introducesthe detail of the algorithm. Section 3 applies our algorithmon sparse LV tMRI and compares it to TPS in running time,deformed shape and triangle quality. In section 4 we draw theconclusions.

2. METHODS

2.1. Framework

Figure 1 shows the framework of our algorithm. The inputis the surface mesh of a generic model, model landmarks andimage landmarks. In our specific case, the generic model issegmented from dense tMRI obtained from a healthy volun-teer and the surface mesh was build by a Delaunay triangula-tion using geodesic distances. The landmarks are calculatedboth on the image contours and on the corresponding slicesof the model. The output is the reconstructed surface mesh.

In between there are several transformation filters. Globaldeformations are used to roughly place the model on the im-age data location based on affine transformation includingtranslation, rotation and isotropic scaling. LSO is employedtwice. The first pass smooths the initial surface mesh. The

Fig. 1. Algorithm framework. Inputs are surface mesh of ageneric model, model landmarks and image landmarks. Out-put is the deformed surface mesh of an image data. In be-tween there are several filters such as global deformation,LSO and LSD.

second pass improves the triangle quality of output. Here tri-angle quality is measured by radius ratio (Section 3). LSDis employed to locally deform the model mesh to image dataaccording to model and image landmarks. Section 2.3 and 2.4give details on these filters.

2.2. Landmarks and Global Deformation

Our previous research [1] describes the detail of obtaininglandmarks and material markers, as well as the data infor-mation. First LV boundaries and tagging line informationare obtained from tMRI using Gabor Filters [9] and Meta-morphs [10]. Secondly, landmarks are evenly selected fromthe boundary lines in short axis (SA). Since there are less then5 slices in SA for sparse data, the landmarks are relativelysparse. Finally, the intersections of the three tagging planesare calculated, as well as the intersections of the LV boundaryand the tagging planes. These intersections can be used as thematerial markers in LV motion tracking.

After obtaining corresponding landmarks from the surfacemesh and tMRI, global deformation is applied by approxi-mately finding the translation, scaling and rotation matrix.The global translation is defined as a vector pointing fromthe center of source points to the center of target points. Theglobal scaling after translation is defined as the ratio betweenthe radius of the source points and the radius of the targetpoints. The global rotation after translation and scaling is ob-tained by Polar decomposition. After applying global trans-lation, scaling and rotation, non-rigid local deformation is re-quired to ensure the accuracy. LSO and LSD are employedhere.

2.3. Laplacian Surface Optimization

Laplacian surface is also called differential coordinates. Itrepresents each point as the difference between such point andits neighborhoods. LSO is an algorithm to improve trianglequality of a surface mesh. The inputs are anchor points andan initial surface mesh. The output is an optimized surfacemesh.

Here we introduce notations and this algorithm. Letthe mesh M be described by a pair (V, E), where V ={v1, ..., vn} describes the geometric positions of the ver-tices in R3 and E describes the connectivity. The neigh-borhood ring of a vertex i is the set of adjacent verticesNi = {j|(i, j) ∈ E} and the degree di of this vertex isthe number of elements in Ni. Instead of using absolutecoordinates V, the mesh geometry is described as a set ofdifferentials ∆ = {δi}. Specifically, coordinate i will berepresented by the difference between vi and the average ofits neighbors:

δi = vi −1di

∑j∈Ni

vj (1)

Assume V is the matrix representation of V. The trans-formation between vertex coordinates V and Laplacian co-ordinates ∆ can be described in matrix algebra. Let Nbe the mesh adjacency (neighborhood) matrix and D =diag(d1, ..., dn) be the degree matrix. Then ∆ = LV , whereL = I −D−1N for the uniform weights.

Using a small subset A ⊂ V of m anchor points, amesh can be reconstructed from connectivity informationalone [4]. x, y and z positions of the reconstructed object(V ′

p = [v′1p, ..., v′np]

T , p ∈ {x, y, z}) can be solved separatelyby minimizing the quadratic energy

‖LV ′p‖2 +

∑a∈A

‖v′ap − vap‖2 (2)

where the vap are anchor (landmark) points. ‖LV ′p‖2 tries

to smooth the object by minimizing the difference, and∑a∈A ‖v′ap − vap‖2 keeps anchor points unchanged. In

practice, with m anchors, the (n + m) × n overdeterminedlinear system AV ′

p = b[LIap

]V ′

p =[

0Vap

](3)

is solved in the least squares sense using the method of normalequations V ′

p = (AT A)−1AT b. The first n rows of AV ′p =

b are the Laplacian constraints, corresponding to ‖LV ′p‖2,

while the last m rows are the positional constraints, corre-sponding to

∑a∈A ‖v′ap − vap‖2. Iap is the index matrix of

Vap, which maps each V ′ap to Vap. The reconstructed shape is

generally smooth, with the possible exception of small areasaround anchor vertices. The minimization procedure moveseach vertex to the centroid of its 1-ring, since the uniformLaplacian L is used, resulting in good inner fairness.

The main computation cost of this algorithm is big matrixmultiplication and inverse. Since A is sparse matrix, AT Ais sparse symmetric definite matrix. The Conjugate Gradientalgorithm can be employed to solve the system.

In our algorithm, LSO is employed twice (Figure 1).At the first time the anchor points are selected evenly andsparsely from the whole object, therefore the shape of surfacemesh of generic model can be roughly conserved as well assmoothed. At the second time image landmarks are used asanchor points, therefore truth points can be fixed and trianglequalities can be improved. The output data are generated afterthe second pass.

2.4. Laplacian Surface Deformation

LSD is also called Laplacian Surface Editing, which is an al-gorithm for local deformation. The inputs are deformed con-trol points and an initial mesh. In our specific case, controlpoints are model and image landmarks. Model landmarks aremoved to image landmarks directly. The deformation of restpoints can be calculated by LSD. Note that after global de-formation process, the displacements of control points are re-stricted in a local range. The output is the deformed mesh.

Using the same notation as Section 2.3, this time we needto minimize this quadratic energy function:

E(V ′) =n∑

i=1

‖δi − δ′i‖2 +∑i∈C

‖vi − v′i‖2 (4)

where δ′ and v′ is Laplacian and Cartesian coordinates afterdeformation. C is the set of control points. The first half tryto keep the shape according to previous time step, which is δ.The second half can move control points v to deformed posi-tions v′. However, using formula 4, no point will be movedexcept the control points C. The main idea of LSD is to com-pute an appropriate transformation Ti for each vertex i whichcan be plugged into energy formula 4:

E(V ′) =n∑

i=1

‖Tiδi − δ′i‖2 +∑i∈C

‖vi − v′i‖2 (5)

Ti should be constrained to avoid a membrane solution,which will lose all geometric detail. Thus, Ti should includerotations, isotropic scales, and translations. In particular,anisotropic scales should not be allowed, as they allow re-moving the normal component from Laplacian coordinates.The class of matrices representing isotropic scales and ro-tation can be written in a specific format. Employing suchformat, formula 5 can be simplified. The detailed formula canbe found in [6]. Formula 5 can be minimized iteratively byfinding Ti and applying it on each vertex coordinates. Whenv converges, this minimization problem is solved. The trans-formation Ti is an approximation of the isotropic scaling androtations when the rotation angle is small. In our framework,

the major rotation is handled in the global deformation part.The local rotation fits the small angle assumption.

After reconstructing the surface of the LV and initializingthe deformation models, [1] [6] [7] [8]can be employed toreconstruct the LV motion.

3. EXPERIMENTS

LSE, LSO and a sparse matrix solver are implemented andtested on a 2.40 GHz Intel Core2 Quad computer with bothGNU/Linux and Windows environments. These algorithmsare coded by extending vtkAlgorithm in Visualization ToolKit(VTK) and named as vtkModeling [11]. Codes and doc-uments can be downloaded in the vtkExtend category ofSourceforge. For comparison, TPS is also tested.

The initial surface mesh of a generic LV model has 2833vertices and 5662 polygons. The image data is MR data withseven SA slices and two LA slices. 270 landmarks are gener-ated from SA image data and surface mesh. The whole 4-filterframework takes about 15 seconds with our matrix solver andless than 1 second with professional solver TAUCS. Note thatthe first pass of LSO can be performed offline, by which thespeed can be improved. TPS takes 3.5 seconds with the samedata sets and professional solver.

Figure 2 shows the effect of each step of our framework.Figure 3 compares results of our algorithm (red object) andTPS (green object). Applying TPS directly on the initial meshresults in bad shape due to its irregular shape and degener-ated triangles. For fairness, we apply TPS on the first passof the LSO. The TPS result is shrunk in regions far awayfrom control points like the bottom, since TPS tries to smooththe whole shape and removes high frequency. Due to non-uniform scaling and shearing, TPS cannot guarantee a rigiddeformation on the top.

Triangle shapes are also measured and compared. Wemeasure our success with the radius ratio mapped to [0, 1] as

ti = 2r

R(6)

where R and r are the radii of the circumscribed and inscribedcircles respectively. This way, ti = 1 indicates a well shapedtriangle, ti ∈ [0, a] means degenerate triangle. Figure 4 com-pares Ti value of our algorithm and TPS. tmean of our meshis 0.8932. tmean of TPS mesh is 0.8628. Note that TPS isapplied on the first output of LSO, whose triangle quality isalready improved.

4. CONCLUSION

We presented a novel framework to reconstruct a surfacemesh from sparse tMRI. The resulting high-quality mesh canbe employed as the input of deformable models to recon-struct the LV motion. We also introduced LSO and LSD tothe medical imaging community. The algorithm is efficient,

Fig. 2. The effects of our algorithm framework showed withpolygon mesh and wire-frame. From left to right, they areinitial surface mesh of a generic LV model, the first LSO filter,LSD filter and the second LSO filter.

Fig. 3. Results from our algorithm (red) and TPS (green)are displayed with tMRI data. TPS result is over-deformedaround control points but under-deformed in other regions.

Fig. 4. Triangle quality measured by Ti histogram. The leftone is obtained from our algorithm. The right one is obtainedfrom TPS.

the resulting mesh represents the image data well, and thetriangle quality is suitable for input of deformable models.Experiments are designed to compare our algorithm to TPS.LSO and LSD are implemented as VTK classes, which canbe easily used in many projects. In the future we will focuson volume data reconstruction.

5. REFERENCES

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